 # Problems On Area And Circumference Of A Circle

Before looking at problems on circle based on perimeter and area, we need to understand both the properties of circle.

## Perimeter Definition

Perimeter is associated with any closed figure like triangle, quadrilateral, polygons or circles. It is the measure of distance covered while going around the closed figure on its boundary.
For example, the perimeter of a square of side 2 cm = 8 cm, as we know that the square comprises 4 sides having equal lengths, thus the total distance covered will be 4×2, which will be the total length (i.e. Perimeter)

## Area Definition

Area means the actual space enclosed by a closed figure (or within the perimeter). It means all the points within the closed figure and not the boundary.

Now, coming to the Perimeter of a circle, as explained above, it is the measure of distance going around the boundary of a circle. This distance is difficult to calculate exactly. Looking at various circles having different radii, it is easy to visualize that the distance to go around is larger in a circle of larger radius than a smaller radius. Hence, the perimeter is a function of the radius of a circle. In case of the circle, we generally use the term “CIRCUMFERENCE” instead of perimeter.

It is given by,

Circumference = 2πr

(Here r is a radius and π is a constant and actually defined as the ratio of circumference to the diameter of a circle).

The value of π is 22/7 or 3.1416

### Solved Problems on Circle

Let us understand the concepts related to circles along with the following questions-

Example 1:

To cover a distance of 100 km a wheel rotates 5000 times. Find the radius of the wheel.

Solution: No. of rotations = 5000.

Total distance covered = 100 km, and we have to find out the radius of the circle.

Let ‘r’ be the radius of the wheel

Circumference of the wheel = Distance covered in 1 rotation = 2πr.

In 5000 rotations, the distance covered = 100 km = 10 x 103x 102 cm.

Hence, in 1 rotation, the distance covered = $\frac{1000000}{5000}cm=200\: cm$

But this is equal to the circumference. Hence, 2πr = 200 cm

r = 200/2π

r = 100/π

Taking the approximate value of π as 22/7, we get

r = 100 x 7/22

r = 31.82 cm approx.

Example 2:

The diameter of the given semi-circular slice of watermelon is 14 cm. What will be the perimeter of the slice of watermelon? Solution: Given diameter=14 cm

Perimeter (p) =2πr

p = 2 x 22/7 x 7= 44 cm.

An interesting fact to note here is that the shape of the slice is semi-circular along with a line.

Thus, perimeter of semi-circular slice is:

$P = \frac{p}{2} + 2r$

$P = \frac{44}{2} + 14$

$P = 36$ cm

Example 3:

The difference between the circumference and the diameter of a circular bangle is 5 cm. Find the radius of the bangle. (Take $\pi = \frac{22}{7}$) Solution: Let the radius of the bangle be ’r’

According to the question:

Circumference – Diameter=5 cm

We know, Circumference of a circle = 2πr

Diameter of a circle = 2r

Therefore, 2πr – 2r =5 cm

2r(π-1) = 5 cm

$2r(\frac{22}{7}-1)=5cm\\ \\ 2r\times \frac{15}{7}=5\\ \\ r=\frac{5\times 7}{15\times 2}\\ \\ r=1.166cm$

The radius of bangle is 1.166 cm.

Example 4:

A girl wants to make a square-shaped figure from a circular wire of radius 49 cm. Determine the sides of a square.

Solution: Let the radius of the circle be ’r’.

Length of the wire=circumference of the circle=$2\pi r$

= $2\times \frac{22}{7}\times 49=2\times 22\times 7\\ \\ =308\: cm$

Let the side of the square be ‘s’.

Perimeter of the square = length of the wire = 4s

$s=\frac{308}{4}\\ \\ s=77\:cm$

Therefore, the sides of the square is 77 cm.